Accessing my jellyfin server locally from my laptop using the server's hostname

[TheDreadPirate]

Any reason why you're against adding a static IP address? For servers, that's the general way to handle it on a local network.

As for using the hostname of your computer, that shouldn't matter for Jellyfin. Just specifying the hostname should be enough to access the computer itself.
Using "hostname : 8096" should be the same as "IP : 8096".

Also as a note, I have found that the Android phone app doesn't work when specifying the hostname. I had to use the IP address I statically assigned to my server for the Android phone app. You may need to do the same for that. I long ago heard an explanation on why this happens, but I don't remember the details as to why this is.

If you've configured your phone to use a private/external DNS, instead of your router, local hostnames won't resolve.

But to the OPs issue, static IPs are preferred for servers.  Not only for simply accessing it at the same IP address, but also because some routers will only accept IPs for port forwarding rules (some accept hostnames).  Assuming you ever plan to make your server remotely accessible.